# Full text of "The Lorentz Invariant Measure and the Heisenberg Uncertainty Principle"

## See other formats

```The Lorentz invariant measure

and the Heisenberg uncertainty principle

BY PATRICK BRUSKIEWICH AND ALICIA KIEtBIK

Abstract

A novel approach to the Heisenberg uncer-
tainty principle is presented in this paper in
a fashion that directly links special relativity and
quantum theory from first principles.

special relativity and
quantum theory

The traditional approach to non-relativistic quan-
tum mechanics is primarily didactic. The basic
axioms of the theory are introduced, along with
the Heisenberg uncertainty principle and simple
systems such as the quantum harmonic oscilla-
tor, are elucidated.

A different approach, which is heuristic in
nature, is to begin with the Lorentz invariant mea-
sure from special relativity. The Lorentz invariant
measure can be used as a conceptual foundation
of quantum mechanics, and a direct link between
special relativity and quantum, mechanics can be
presented without recourse to subsidiary con-
ditions. As a result of this heuristic approach, a
foundation can be laid that firmly joins the two
fields, which are typically presented separately,
until second quantization and the Klein-Gordon
and Dirac equations are presented.

The Lorentz invariant
measure

As outlined, in this paper, it appears evident thai
special relativity and quantum mechanics can be
directly linked from first principles. Starting with
the Lorentz invariant measure, a novel approach to
the Heisenberg uncertainty principle is presented.
The Lorentz invariant measure is 1

2 2 2 2 / 2 1 2 \

ex =c r -lx +y +z \,

(1)

which joins two points in Minkowski space. In
this paper, we can look at the xt-plane without
any loss of generality. For a one-dimensional
inertial system,

2 2 2^2 2 f-)\

C T =C t~ -x . U)

Multiplying through by mass m, and rear-
ranging terms, we get

2 2.2 22 / n \

mx -mc i = -mc x . (3)

Take the partial derivative of this expression
with respect to X, the proper time ("temps proper"
which literally means one's own time) and sim-
pl i fy. This yields

dx 2 dt

xm tmc — = —mc " x.

dx dx

14)

From special relativity, m(dx/dx) = my(3x/9t) =
ymv = p, and (dt/dx) = y, so then

xp ~Et = -mc~% (5)

is also invariant. In 3D, this is the familiar inner
product of the two four- vectors, namely

x Pm p =-mc 1 x. (6)

In quantum mechanics, this measure is impor-
tant when describing a plane wave (unction

\|/ =exp

i(px—Et)

ft

= exp

i*-p>)

(7!

or a wave packet made up of a superposition of
states such as

j/K/?)exp

i^p»)

d N p, (8)

w T here A(p) is a momentum dependent amplitude,
and N is the dimension of the space.
By inspection, it is evident, then, that

x P,

(9)

represents a phase (p.

Measurement in quantum mechanics involves
both amplitude and phase considerations. For
information to be pass between two events in
space-time a signal must be exchanged between
the two events. The fastest signal that can be
exchanged travels at the speed of light. As a signal,
or wave, travels between two points in space, its
change in phase may be used as a measure of the
separation between the two points in space-time.

Phase difference
in Minkowski space-time

In studying quantum measures, we can use dif-
ferent approaches, such as the limiting process
and infinitesimal analysis, a technique developed
at Cambridge beginning in Newton's time. In
this paper, we shall use. infinitesimal analysis. 2

In recent times, some of the infinitesimal analy-
sis techniques have also been used in Wiener
measures and Ito calculus in stochastic quantum
mechanics.

In ID, the difference in phase between two
adjacent events in Minkowski space-time, two
points infinitesimally separated by a causal
connection (i.e. two points on or within the light
cone connecting the two events) is 3

5(p « <f> 2 -<p, = (x 2 p 2 - E 2 t 2 ) - (x, p } - E,t\ )
= (x l +§x)(p l +dp)-{E l +bE)(t l +8t)

= bxp { +xfip+ 8x8p - 3Et } - E { 8t - 8E8t.

(10)
This expression can be grouped into three
terms First, consi der the bEt } term. Given thai
E - • N /pV+ m V , then

5£ - 1 5p = v 5p - _J- qV,

E t

(ID

where we have used p = ymv and E = ymc 2 . Since

v = 5x/5t, this means vt, = x } . So for two points

infinitesimally separated by a causal connection,

X[8p - 5Et[ - and these two terms cancel each

other out. in the expression.

Consider the E,or term:

( -< 2 ^
£,5/ = ymc 2 3t = (ymv ) — 6/ = pfix,

\ i J

(12)
where we have used the group velocity v g given
by v = c 2 /v g . For a wave disturbance, whether it is
a massless particle like a photon or a massive par-
ticle like an electron, it is the group velocity that
characterizes the momentum or energy o{ the
w T ave packet, and therefore the exchange of infor-
mation between two points in space-time. For
two points infinitesimally separated by a causal
connection, Efit - p { 3x = and these two terms
cancel each other out in the expression.
Of the six terms in the infinitesimal phase
expression, only two terms remain: the differ-
ence in phase between two points infinitesimally
separated by a causal connection in space-time
given by

8(p~5x6>-8£5/. (13)

From the standpoint of infinitesimals, this
expression is counter-intuitive in that the phase
is made up of a product of terms and we would
expect an infinitesimal change in the phase to be
first order in these terms. The fact the first-order

Volume VI Issue 3 APRlt2008

infinitesimal terms drop out of the expression
is a direct result of the Lorentz invariant mea-
sure and appears to form the conceptual basis
to the Heisenberg uncertainty principle. It is at
this juncture that special relativity and quantum
mechanics are linked at first principles.

For a massless particle like a photon, then

8E - cSp => 8Ebt = Spcdt = dpdx. (14)

As is evident, near an extremum where Sep is
small, then

dEdt « 8p8x.

(15)

Based on our derivation from the invariant
measure, there is nothing that predisposes that
Sep = 0, 8E8t ~ 0, or 8p8x ~ 0. When we consider
that information exchange requires the exchange
of energy or momentum, then by dimensional
analysis, we find

8E8t « 5>Sx ~ h. (16)

The classical standpoint

Let us consider this expression from a classical
standpoint, in terms of generalized momentum
p and generalized q. By the equivalence of work-
energy, we have

\dE=\Fdq=\^d q .

(17)

We consider the phase space of a quasi-sta-
tionary or periodic system and take a time aver-
age

{dp .

J**-* J**-

(18)

cance of this remarkable constant." 1

The Dfrac delta potential

It is straightforward to use this quantum action to
solve for the energy states of a particle in a box or
for a quantum harmonic oscillator. A more chal-
lenging system to studying using the phase space
approach is that of a particle trapped by a Dirac
delta potential.

Consider a potential of finite size and shrink
the region of interaction to a point. Expressed as
a ID attractive Dirac delta potential of strength a
the Schrodinger equation for this system is

ft 2 d 2
- \|/-a5(x)y = £\|/. (21)

2m ax

There are a number of ways to solve this second
order di fferential equation, including using integral
transforms. The integral transform methods are
outlined in a paper titled "A Novel Look at the One
Dimensional Delta Schrodinger Equation/' written
by one of the authors for the Gamma Magazine of
the Niels Bohr Institute in May 2005. 5

As is well known, this equation has the single
solution

i|/(x) = \|/(0)exp

By inspection,
px ma\x\

-ma x

ma

(22)

(23)

ft ft 2 n

and so we can solve for the energy of this state,
namely

N-

di J 5/

This implies then that

5/ jdE » jdpdq ^> StdE w dpdq. (19)

The classical concept of phase space is consis-
tent with the derivative from the Lorentz invari-
ant measure.

At the Solvay Conference in 1911, Max Planck
introduced the quantum action, namely

jdpdq = nth (20)

where n is an integer and ti is Planck's constant.
This phase integral represents the quantization of
quantum action. Planck expressed the view that
"one should therefore confine oneself to the prin-
ciple that the elementary region of probability, ft 8p8x
has an ascertainable finite value and avoid any
further speculation about the physical signifi-

WWW.CUPJ.CA

{ma I h)

tmx^ (24)

2m 2m 2ft 2

The velocity of the particle can be derived

(25)

from

ma a 8x

p = mv = => v - — = — .

ti ft 8t

'fake the derivative of the momentum with
respect to the potential strength a:

8am

8p = -

(26)

if we multiply both sides of this equation with
ox = abl/ft, then

_( 8am Va5/

"I » A »

(27)

\

We see that while we are dealing with a
delta potential, what this last expression implies
is that neither 5p6x — » nor 8E8t —> for this
system. Infinitesimal analysis of this system
shows that

5£5/ a 6>5x « ft, (28)

as expected.

Conclusion

The Lorentz invariant measure can be used as a
conceptual foundation of quantum mechanics,
and a direct link between special relativity and
quantum mechanics can be presented without
recourse to subsidiary conditions. In this paper,
using infinitesimal analysis, a technique devel-
oped at Cambridge beginning in Newton's time,
we have derived the Heisenberg uncertainty
principle from first principles.

Acknowledgements

Patrick would like to thank Dr. J. McKenna and
Dr. M. McMillan for their valuable insights on
the Lorentz invariant measure, and both Dr. F.
Kaempffer and Dr. A. Zhitnisky for their unique
insights on the Heisenberg uncertainty prin-
ciple.

References

] C. Moller. The Theory of Relativity, Oxford Uni-
versity Press. 1972, p. 97.

2 J.L. Bell. A Primer of Infinitesimal Analysis. Cam-
bridge University Press: Cambridge. 1998,
p. 29.

3 R. Schlegel. Superposition and Interaction. Univer-
sity of Chicago Press: Chicago. 1980, p. 198.

l M. Jammer. The Conceptual Development of
Quantum Mechanics. McGraw Hill: New York.
1966, p. 54.

5 P. Bruskiewich. "A Novel Look at the One
Dimensional Delta Schrodinger Equation."
Gamma Magazine. 138 (May 2005).

Alicja Kielbik is a third year undergrad at the
University of British Columbia (UBC). She is
interested in quantum continuous measurement
theory and its application to quantum computing
and gravity.

Patrick Bruskiewich is a UBC doctoral can-
didate and editor-in-chief of CUPj. He can be
reached at patrichh@phas.uhc.ca.

*